Non-Lorentzian line-shape

The NMR provided invaluable information on the structures of a variety of assemblies of 6-quartets. In the presence of Na and K. an octameric structure of 5 -CMP is formed in water.The analysis of the non-Lorentzian line shapes of the Na-NMR signals afforded a measurement of the correlation time of the reorienting, rigid, supramolecular entity, confirming the size of the aggregate, in complement to the H- and - P-NMR structural data. ... 

Figure 10 shows linearized DISPA plots for two simulated non-Lorentzian line shapes. For example. Figure 10a shows the DISPA deviations for a line shape consisting of an unresolved doublet of two Lorentzians of different resonant frequency. The maximum (upward) displacement from the reference line clearly occurs at frequencies less than one half-width away from the observed absorption maximum. In contrast. Figure 10b shows that for a log-Gauss distribution in relaxation time, the maximum displacement (downward this time) occurs at a frequency approximately equal to the half-width at half-maximum height of the observed absorption peak. 

The excellent agreement between the experimental data and the reference semicircle expected for a single Lorentzian line in Figure 11 shows that any deviations from the reference circle for subsequent data sets will reflect non-Lorentzian line shape from the sample rather than artifacts from the instrument or the data reduction. 

Figure 2.5 shows a few of the Lorentzian line shapes in the Gaussian distribution of the real part (top) and imaginary part (bottom) of the resonant non-linear susceptibility... 

In indirect methods, the resonance parameters are determined from the energy dependence of the absorption spectrum. An important extra step — the non-linear fit of (t E) to a Lorentzian line shape — is required, in addition to the extensive dynamical calculations. The procedure is flawless for isolated resonances, especially if the harmonic inversion algorithms are employed, but the uncertainty of the fit grows as the resonances broaden, start to overlap and melt into the unresolved spectral background. The unimolecular dissociations of most molecules with a deep potential well feature overlapping resonances [133]. It is desirable, therefore, to have robust computational approaches which yield resonance parameters and wave functions without an intermediate fitting procedure, irrespective of whether the resonances are narrow or broad, overlapped or isolated. 

Numerical problems arising from the use of the Lorentzian for fitting spectra have also been reported [69]. These are related to the non-dif-ferentiable profile at the points where the exponential wings are attached to the Lorentzian core. Partial derivatives with respect to the line shape parameters are usually needed in least mean squares fitting procedures. 

A pseudo solid-like behavior of the T2 relaxation is also observed in i) high Mn fractionated linear polydimethylsiloxanes (PDMS), ii) crosslinked PDMS networks, with a single FID and the line shape follows the Weibull function (p = 1.5)88> and iii) in uncrosslinked c/.s-polyisoprenes with Mn > 30000, when the presence of entanglements produces a transient network structure. Irradiation crosslinking of polyisoprenes having smaller Mn leads to a similar effect91 . The non-Lorentzian free-induction decay can be a consequence of a) anisotropic molecular motion or b) residual dipolar interactions in the viscoelastic state. 

FIGURE 8.5 Non-Lorentzian 23Na line shape from a solution of NaCl and malonyl gramicidin in lysolecithin micelles, (a) Experimental spectrum. (f>) Computer-generated Lorentizian components. FromVenkatachalam and Urry.92... 

Another technique for separating different contributions to the FID which has been used sucessfully is to simply drop off points at the beginning of the FID (Roth, 1980). Suppose the NMR line consists of a Lorentzian portion and a broader non-Lorentzian portion which we can suppose to be Gaussian without altering the argument. As we discussed in the last section, the Lorentzian line gives rise to an exponential FID for which the shape is independent of the portion of the signal considered. 

The first difference with incoherent scattering is that the intensity varies as S(Q), leaving aside the influence of the Debye-Waller factor. One, therefore, expects a non-monotonic variation of the total intensity as a fimction of Q. In the small Q domain, the line-shape of the coherent scattering fimction is still Lorentzian, with a HWHM... 

If the theoretical line shape is known for a peak, the function can be fitted to the data, and the intensity, width, and integral determined. In practice, however, experimental line shapes can be too complex to characterize conveniently. NMR lines, for example, have been shown by the Bloch equations to be Lorentzian in shape. A number of experimental factors, however, can contribute non-Lorent-zian components to the observed line shape. Furthermore, unresolved peaks will have to be fitted to a sum of individual Lorentzians rather than to a single curve. 

Figure 2 shows that a single Lorentzian line yields a DISPA "reference" circle. For a DISPA plot for any other experimental line shape, any deviation from a reference circle having the same absorption peak height will then reflect non-Lorentzian composite line shape. We will next examine DISPA plots for several types of linebroadening, to determine the direction and magnitude of the displacement from the corresponding "reference" circle. 

The line shapes of two-photon transitions are very simple as they are simply lorentzian curves, whereas the line shape in the saturation technique is quite complicated (its calculation involves the averaging of a non-linear effect which depends on the velocity component v ). In case of collisions, the two-photon line shape remains a lorentziai one and it is easy to measure the broadening and the shifts, whereas the velocity-changing collisions complicate still further the already complicated line shapes of saturation spectroscopy. 

The question now arises as to what is meant by a resonant intermediate state. Spectral lines are generally Lorentzian in shape thus, as we move away from the line centre, the intensity drops quite rapidly but remains non-zero for a considerable distance from the line centre. Excitation in the wings of a line is still possible if the laser intensity is high enough. However, at high intensities, coherent excitation processes become important, and two or more photons can be absorbed simultaneously through virtual states (i.e. in the absence of real/resonant intermediate states). Theory shows that such transitions result from the collective effect of all allowed transitions, inversely weighted by the difference in... 

We can also relate these two approximations through the stochastic theory of the line shape developed by Kubo [11] and applied to molecular line shapes by Saven and Skinner [10]. As shown by Kubo, the overlap function given by Eq. (13) is a general result for a Gaussian-dis-tributed random variable in a Markovian process [11]. In the limit of a very slow decay of the time-correlation function of this random variable, the overlap function reduces to Eq. (9) and the line has a Lorentzian shape. In the limit of a very fast decay of the time-correlation function, the overlap function reduces to Eq. (15) and the line has a Gaussian shape. Employing molecidar dynamics simulations of chromophores within non-polar fluids. 

This equation holds only for an optical electric field which is Gaussian and which possesses an exponential autocorrelatitxi function, i.e., Eq. (66). If the field is non-Gaussian there is no simple relation between the optical spectrum lio)) and the power spectrum. Eq. (69) has three components (1) a shot noise term e(S ln which is independoit of the frequency (Le., white noise), (2) a d.a photocurrent 6(m) which is essentially infinite at extremely low fiequendes (i.e., d.c.) and a light beating spectrum which for an exponential autocorrelation function and Gaussian optical field is a Lorentzian of half width, IF. Fig. 4 shows the experimental data of Benedek et al with calculated points and observed line shape. What is not shown is the infinite d.c. photocurrent at ft)=0. These measurements were obtained by use of a spectrum analyzer which measures directly the power scattered at each frequency. 

It is interesting to note that the analysis of the line shape mentioned above cannot be applied to non-stoichiometric YFe204 at low temperatures. The saw-tooth profile of the diffraction peak cannot be interpreted well by an assumption of a Lorentzian or Gaussian function for the intensity profile perpendicular to the c -axis but is intermediate between them. The spin system is not... 

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